A Statistician collected the data of 78 values with two independent variable X₁ and X₂, and considered the four models: (i) Y = B₀ + e;

(ii) Y = B₀ + B₁X₁ + e;

(iii) Y = B₀ + B₁ X₁ + e and

(iv) Y = B₀ + B₁ X₁ + B₂ X₂ + e.

The results obtained are: σ²=0.91, SS(B₀) = 652.42, SS(B₀,B₁) = 679.34, SS(B₀,B₂) = 654.00,SS(B₀,B₁,B₂) = 687.79 . Find the additional contribution of (i) X₂ over X₁ and (ii) X₁ over X₂. Test whether their inclusion in the model is justified.

The following data comprising the number of customers (in hundred) and monthly sales (in thousand Rupees): 

 Calculate the residuals and determine the standardised residuals for the model Y = 2.6185 + 0.4369 X 

Use graphical method to minimise the time added to process the following jobs on the machines shown: 

 Calculate the total time elapsed to complete both the jobs.

A department head has four subordinates, and four tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each man would take to perform each task, is given in the table below:

How should the tasks be allocated, one to a subordinate, so as to minimise the total man hour?

Solve the following LPP using graphical method:
Maximize Z = 3x₁ + 2x₂
Subject to the Constraints:
 -2x₁ + x₂ ≤ 1
 x₁ ≤ 2
 x₁ + x₂ ≤ 3
 x₁, x₂ ≥0 

Rewrite the following linear programming problem in Standard form:
Minimise Z = 2x₁ + x₂ + 4x₃
Subject to the Constraints:
 -2x₁ + 4x₂ ≤ 4
 x₁ +2x₂ + x₃ ≥ 5
 2x₁ + 3x₃ ≤ 2
 x₁ ≥ 0, x₂ ≥0, x₃ ≥ 0 

State whether the following statements are True or False and also give the reason in support of your answer.
a) The Set $ ={(x, y) : 0 ≤ y ≤ 5 when 0 ≤ x≤ 2 and 3≤ y≤ 5 when 2≤ x ≤ 7 } is not a convex set.
b) If 10 is added to each of the entries of the cost matrix of a 3 x 3 assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
c) The solution to a transportation problem with m-rows (supplies) and n-columns (destinations) is feasible if number of positive allocations is m + n.
d) The Value di ≥ 3 indicates an outlying observation in regression analysis.
e) Variations which occur due to natural forces and operate in a regular and periodic manner over a span of less than or equal to one year are termed as cyclic variations

The failure data of 10 electronic components are shown in the table given below:

Estimate, the
i) reliability. 
ii) cumulative failure distribution.
iii) failure density. 
iv) failure rate functions.

A small electronic device is designed to emit a timing signal of 200 milliseconds (ms) duration. In the production of this device, 10 subgroups of four units are taken at periodic intervals and tested. The results are shown in the following table: 

 i) Estimate the process mean and standard deviation. 
ii) Determine the centre line and control limits for the process mean and process variability.
iii) By plotting the charts on graph paper, determine that the process is stable or not with respect to the process mean and process variability. If necessary, compute revised control limits

The system shown below is made up of ten components. Components 3, 4 and 5 are not identical and at least one component of this group must be available for system success. Components 8, 9 and 10 are identical and for this particular group it is necessary that two out of the three components functions

Solve the two-person zero-sum game having the following payoff matrix for player A

The failure density function of a random variable T is given by

Calculate, the
i) reliability of the component. 
ii) reliability of the component for a 100 hour mission time. 
iii) mean time to failure (MTTF). 
iv) median of the random variable T. 
v) life of the component, if the reliability of 0.96 is desired. 

A shirt manufacturing company supplies shirts in lots of size 250 to the buyer. A single sampling plan with n = 20 and c = 1 is being used for the lot inspection. The company and the buyer decide that AQL = 0.04 and LTPD = 0.10. If there are 15  defective in each lot, compute the
i) probability of accepting the lot. 
ii) producer’s risk and consumer’s risk. 
iii) average outgoing quality (AOQ), if the rejected lots are screened and all defective shirts
are replaced by non-defectives. 
iv) average total inspection (ATI). 

To monitor the manufacturing process of mobile phones, a quality controller randomly selected 100 mobile phones from the production line, each day over 15 days. The mobile phones were inspected for defectives and the number of defective mobile phones found each day was recorded. The data are given below:

 i) Determine the trial centre line and control limits for the fraction defective using the above data. 

ii) Contract the control chart on graph paper and determine that the process is stable or not. If there is any out-of-control point, determine the revised centre line and control limits.

State whether the following statements are True or False. Give reason in support of your answer. 
a) If the average number of defects in an item is 4, the upper control limit of the c-chart will be 12.
b) The specification limits and natural tolerance limits are same in statistical quality control.
c) If the probability of making a decision about acceptance or rejection of a lot on the first sample is 0.80 and the sizes of the first and second samples are 10 and 15, respectively, then
the average sample number for the double sampling plan will be 25.
d) Two independent components of a system are connected in series configuration. If the reliabilities of these components are 0.1 and 0.30, respectively then the reliability of the system will be 0.65.
e) A point in the pictorial representation of a decision tree having states of nature as immediate sub-branches is known as decision point. 

State whether the following statements are True or False. Give reason in support of your answer. 
a) If the average number of defects in an item is 4, the upper control limit of the c-chart will be 12.
b) The specification limits and natural tolerance limits are same in statistical quality control.
c) If the probability of making a decision about acceptance or rejection of a lot on the first sample is 0.80 and the sizes of the first and second samples are 10 and 15, respectively, then
the average sample number for the double sampling plan will be 25.
d) Two independent components of a system are connected in series configuration. If the reliabilities of these components are 0.1 and 0.30, respectively then the reliability of the system will be 0.65.
e) A point in the pictorial representation of a decision tree having states of nature as immediate sub-branches is known as decision point. 

Write an essay on British Drama in the twentieth Century.

Comment on the historical significance of Look Back in Anger. 

Discuss Murder in the Cathedral as a poetic drama. 

Discuss the art of characterisation in The Playboy of the Western World?